Recent content by Orodruin

Hint: This is one of those cases where “doing the infinite sum” is a bad bad idea. You might technically be able to do it that way, but it really helps to stop and think for a bit. This problem is a real classic.

Wordle 1 075 4/6 ⬜⬜⬜⬜⬜ ⬜⬜⬜🟩🟩 ⬜🟩⬜🟩🟩 🟩🟩🟩🟩🟩 I liked how this one developed with my attempts. It also caps off the 400th day in a row that I play despite lots of tumultuous happenings in my life. Edit: Play successfully I should add. My laat streak was cut short by … human error. I had only one...

Use that the result of the angular integral must be isotropic. An isotropic tensor of rank 2n can be written as a multiple of the fully symmetrized products of n Kronecker deltas. Edit: I think I spoke a bit fast there. Of course, with the Kronecker delta being isotropic, so will any product of...

What happens with the integrand if you let ##x_i \to -x_i##?

I suggest that you define ##k = x/y## (the case ##y = 0## trivially gives ##a > 0## as a requirement) and require that ##V(ky,y) \geq 0## regardless of ##k##.

To add to the above, the entire point of that problem is to demonstrate how Taylor's theorem works. Yes, it is a known function and you can compute the error exactly, but that is not the point. It is a good and instructive idea to use Taylor's theorem to estimate the error and then compare to...

Yes. This is the definition of a normal subgroup. That this holds regardless of what ##r## is.

I think I intended to write out ”can be no function”, but yes, the same idea. Fixed it. Edit: Indeed, more hands-on even, if the factor in front of ##dt## is ##1/\sqrt{\tilde x}## then ##\partial \tilde t/\partial t = 1/\sqrt{\tilde x}## implying that ##\tilde t = t/\sqrt{\tilde x} + f(\tilde...

In fact, if R is the subgroup of rotations about a specific point, you must take r’ = r.

Yes.

Apart from what was already said, what kind of coordinate transformation do you imagine that would satisfy this? Coordinate differentials by their nature as the differential of coordinate functions must be exact. The RHS here is not exact and therefore there can be no function ##\tilde...

The solution is always yet another hour+ video ... 😏

Wordle 1 073 3/6 🟩⬜🟩⬜🟨 🟩⬜🟩🟩⬜ 🟩🟩🟩🟩🟩 One of the few times I only used one of my two seed words. Unfortunately didn’t mean getting it in two